How can I smooth the top side of this ring?
I guess you have these 2 solutions:


Smooth your object in the Tools panel, and in the Properties panel > Data > Normals, enable Auto-Smooth, set the value under which the angles smooth.





Create a bevel all along the angles, thin enough to keep them sharp, it may need a bit of knife work if you have triangles or ngons. After that, smooth your object in the Tools panel, and give it a Subdivision Surface modifier. If you're supposed to 3D print the object, I guess you just need to apply the Subsurf.